An Introduction to Bioinformatics
2. Comparing biological sequences:
sequence alignment (cont’d)
An Introduction to Bioinformatics
Outline
•
•
•
•
Global Alignment
Scoring Matrices
Local Alignment
Alignment with Affine Gap Penalties
An Introduction to Bioinformatics
From LCS to Alignment: Change up the Scoring
• The Longest Common Subsequence (LCS) problem—the
simplest form of sequence alignment – allows only insertions
and deletions (no mismatches).
• In the LCS Problem, we scored 1 for matches and 0 for indels
• Consider penalizing indels and mismatches with negative
scores
• Simplest scoring schema:
+1 : match premium
-μ : mismatch penalty
-σ : indel penalty
An Introduction to Bioinformatics
Simple Scoring
• When mismatches are penalized by –μ,
indels are penalized by –σ,
and matches are rewarded with +1,
the resulting score is:
#matches – μ(#mismatches) – σ (#indels)
An Introduction to Bioinformatics
The Global Alignment Problem
Find the best alignment between two strings under a given
scoring schema
Input : Strings v and w and a scoring schema
Output : Alignment of maximum score
↑→ = -б
= 1 if match
= -µ if mismatch
si,j = max
si-1,j-1 +1 if vi = wj
s i-1,j-1 -µ if vi ≠ wj
s i-1,j - σ
s i,j-1 - σ
m : mismatch penalty
σ
: indel penalty
An Introduction to Bioinformatics
Scoring Matrices
To generalize scoring, consider a (4+1) x(4+1) scoring
matrix δ.
In the case of an amino acid sequence alignment, the
scoring matrix would be a (20+1)x(20+1) size. The
addition of 1 is to include the score for comparison
of a gap character “-”.
This will simplify the algorithm as follows:
si-1,j-1 + δ (vi, wj)
si,j = max
s i-1,j + δ (vi, -)
s i,j-1 + δ (-, wj)
An Introduction to Bioinformatics
Measuring Similarity
• Measuring the extent of similarity between
two sequences
• Based on percent sequence identity
• Based on conservation
An Introduction to Bioinformatics
Percent Sequence Identity
• The extent to which two nucleotide or amino
acid sequences are invariant
AC C TG A G – AG
AC G TG – G C AG
mismatch
indel
70% identical
An Introduction to Bioinformatics
Making a Scoring Matrix
• Scoring matrices are created based on
biological evidence.
• Alignments can be thought of as two
sequences that differ due to mutations.
• Some of these mutations have little effect on
the protein’s function, therefore some
penalties, δ(vi , wj), will be less harsh than
others.
An Introduction to Bioinformatics
Scoring Matrix: Example
A
R
N
K
A
5
-2
-1
-1
R
-
7
-1
3
N
-
-
7
0
K
-
-
-
6
• Notice that although R
and K are different amino
acids, they have a
positive score.
• Why? They are both
positively charged amino
acids will not greatly
change function of
protein.
An Introduction to Bioinformatics
Conservation
• Amino acid changes that tend to preserve the
physico-chemical properties of the original
residue
• Polar to polar
• aspartate  glutamate
• Nonpolar to nonpolar
• alanine  valine
• Similarly behaving residues
• leucine to isoleucine
An Introduction to Bioinformatics
Scoring matrices
• Amino acid substitution matrices
• PAM
• BLOSUM
• DNA substitution matrices
• DNA is less conserved than protein
sequences
• Less effective to compare coding regions at
nucleotide level
An Introduction to Bioinformatics
PAM
• Point Accepted Mutation (Dayhoff et al.)
• 1 PAM = PAM1 = 1% average change of all amino
acid positions
• After 100 PAMs of evolution, not every residue will
have changed
• some residues may have mutated several
times
• some residues may have returned to their
original state
• some residues may not changed at all
An Introduction to Bioinformatics
PAMX
• PAMx = PAM1x
• PAM250 = PAM1250
• PAM250 is a widely used scoring matrix:
Ala
Arg
Asn
Asp
Cys
Gln
...
Trp
Tyr
Val
A
R
N
D
C
Q
Ala
A
13
3
4
5
2
3
Arg
R
6
17
4
4
1
5
Asn
N
9
4
6
8
1
5
Asp
D
9
3
7
11
1
6
Cys
C
5
2
2
1
52
1
Gln
Q
8
5
5
7
1
10
Glu
E
9
3
6
10
1
7
Gly
G
12
2
4
5
2
3
His
H
6
6
6
6
2
7
Ile
I
8
3
3
3
2
2
W
Y
V
0
1
7
2
1
4
0
2
4
0
1
4
0
3
4
0
1
4
0
1
4
0
1
4
1
3
5
0
2
4
Leu
L
6
2
2
2
1
3
Lys ...
K ...
7 ...
9
5
5
1
5
1
2
15
0
1
10
An Introduction to Bioinformatics
BLOSUM
• Blocks Substitution Matrix
• Scores derived from observations of the
frequencies of substitutions in blocks of
local alignments in related proteins
• Matrix name indicates evolutionary distance
• BLOSUM62 was created using sequences
sharing no more than 62% identity
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The Blosum50 Scoring Matrix
An Introduction to Bioinformatics
Local vs. Global Alignment
• The Global Alignment Problem tries to find
the longest path between vertices (0,0) and
(n,m) in the edit graph.
• The Local Alignment Problem tries to find the
longest path among paths between arbitrary
vertices (i,j) and (i’, j’) in the edit graph.
An Introduction to Bioinformatics
Local vs. Global Alignment
• The Global Alignment Problem tries to find the
longest path between vertices (0,0) and (n,m) in the
edit graph.
• The Local Alignment Problem tries to find the
longest path among paths between arbitrary
vertices (i,j) and (i’, j’) in the edit graph.
• In the edit graph with negatively-scored edges,
Local Alignmet may score higher than Global
Alignment
An Introduction to Bioinformatics
Local vs. Global Alignment (cont’d)
• Global Alignment
--T—-CC-C-AGT—-TATGT-CAGGGGACACG—A-GCATGCAGA-GAC
| || | || | | | |||
|| | | | | ||||
|
AATTGCCGCC-GTCGT-T-TTCAG----CA-GTTATG—T-CAGAT--C
• Local Alignment—better alignment to find
conserved segment
tccCAGTTATGTCAGgggacacgagcatgcagagac
||||||||||||
aattgccgccgtcgttttcagCAGTTATGTCAGatc
An Introduction to Bioinformatics
Local Alignment: Example
Local alignment
Global alignment
Compute a “mini”
Global Alignment to
get Local
An Introduction to Bioinformatics
Local Alignments: Why?
• Two genes in different species may be similar over
short conserved regions and dissimilar over
remaining regions.
• Example:
• Homeobox genes have a short region called the
homeodomain that is highly conserved between
species.
• A global alignment would not find the
homeodomain because it would try to align the
ENTIRE sequence
An Introduction to Bioinformatics
The Local Alignment Problem
• Goal: Find the best local alignment between
two strings
• Input : Strings v, w and scoring matrix δ
• Output : Alignment of substrings of v and w
whose alignment score is maximum among
all possible alignment of all possible
substrings
An Introduction to Bioinformatics
The Problem with this Problem
• Long run time O(n4):
- In the grid of size n x n there are ~n2
vertices (i,j) that may serve as a source.
- For each such vertex computing alignments
from (i,j) to (i’,j’) takes O(n2) time.
• This can be remedied by giving free rides
An Introduction to Bioinformatics
Local Alignment: Example
Local alignment
Global alignment
Compute a “mini”
Global Alignment to
get Local
An Introduction to Bioinformatics
Local Alignment: Example
An Introduction to Bioinformatics
Local Alignment: Example
An Introduction to Bioinformatics
Local Alignment: Example
An Introduction to Bioinformatics
Local Alignment: Example
An Introduction to Bioinformatics
Local Alignment: Example
An Introduction to Bioinformatics
Local Alignment: Running Time
• Long run time O(n4):
- In the grid of size n x n
there are ~n2 vertices (i,j)
that may serve as a
source.
- For each such vertex
computing alignments
from (i,j) to (i’,j’) takes
O(n2) time.
• This can be remedied by
giving free rides
An Introduction to Bioinformatics
Local Alignment: Free Rides
Yeah, a free ride!
Vertex (0,0)
The dashed edges represent the free rides from
(0,0) to every other node.
An Introduction to Bioinformatics
The Local Alignment Recurrence
• The largest value of si,j over the whole edit graph is the
score of the best local alignment.
• The recurrence:
0
si,j = max
si-1,j-1 + δ (vi, wj)
s i-1,j + δ (vi, -)
s i,j-1 + δ (-, wj)
Notice there is only
this change from the
original recurrence of
a Global Alignment
An Introduction to Bioinformatics
The Local Alignment Recurrence
• The largest value of si,j over the whole edit graph is the
score of the best local alignment.
• The recurrence:
0
si,j = max
si-1,j-1 + δ (vi, wj)
s i-1,j + δ (vi, -)
s i,j-1 + δ (-, wj)
Power of ZERO: there is
only this change from the
original recurrence of a
Global Alignment - since
there is only one “free ride”
edge entering into every
vertex
An Introduction to Bioinformatics
The Smith-Waterman algorithm
Idea: Ignore badly aligning regions
Modifications to Needleman-Wunsch:
Initialization: F(0, j) = F(i, 0) = 0
Iteration:
F(i, j) = max
0
F(i – 1, j) – d
F(i, j – 1) – d
F(i – 1, j – 1) + s(xi, yj)
An Introduction to Bioinformatics
Scoring Indels: Naive Approach
• A fixed penalty σ is given to every indel:
• -σ for 1 indel,
• -2σ for 2 consecutive indels
• -3σ for 3 consecutive indels, etc.
Can be too severe penalty for a series of
100 consecutive indels
An Introduction to Bioinformatics
Affine Gap Penalties
• In nature, a series of k indels often come as a
single event rather than a series of k single
nucleotide events:
This is more
likely.
Normal scoring would
give the same score This is less
for both alignments
likely.
An Introduction to Bioinformatics
Accounting for Gaps
• Gaps- contiguous sequence of spaces in one of the
rows
• Score for a gap of length x is:
-(ρ + σx)
where ρ >0 is the penalty for introducing a gap:
gap opening penalty
ρ will be large relative to σ:
gap extension penalty
because you do not want to add too much of a
penalty for extending the gap.
An Introduction to Bioinformatics
Affine Gap Penalties
• Gap penalties:
• -ρ-σ when there is 1 indel
• -ρ-2σ when there are 2 indels
• -ρ-3σ when there are 3 indels, etc.
• -ρ- x·σ (-gap opening - x gap extensions)
• Somehow reduced penalties (as compared to
naïve scoring) are given to runs of horizontal
and vertical edges
An Introduction to Bioinformatics
Compromise: affine gaps
Gap cost
σ
ρ
An Introduction to Bioinformatics
Affine Gap Penalties and Edit Graph
To reflect affine gap
penalties we have to
add “long” horizontal
and vertical edges to
the edit graph. Each
such edge of length x
should have weight
- - x *
An Introduction to Bioinformatics
Adding “Affine Penalty” Edges to the Edit Graph
There are many such edges!
Adding them to the graph
increases the running time
of the alignment algorithm
by a factor of n (where n is
the number of vertices)
So the complexity increases
from O(n2) to O(n3)
An Introduction to Bioinformatics
Manhattan in 3 Layers
ρ
δ
δ
σ
δ
ρ
δ
δ
σ
An Introduction to Bioinformatics
Affine Gap Penalties and 3 Layer Manhattan Grid
• The three recurrences for the scoring
algorithm creates a 3-layered graph.
• The top level creates/extends gaps in the
sequence w.
• The bottom level creates/extends gaps in
sequence v.
• The middle level extends matches and
mismatches.
An Introduction to Bioinformatics
Switching between 3 Layers
• Levels:
• The main level is for diagonal edges
• The lower level is for horizontal edges
• The upper level is for vertical edges
• A jumping penalty is assigned to moving from the
main level to either the upper level or the lower level
(-- )
• There is a gap extension penalty for each
continuation on a level other than the main level (-)
An Introduction to Bioinformatics
The 3-leveled Manhattan Grid
An Introduction to Bioinformatics
Affine Gap Penalty Recurrences
si,j =
max
s i-1,j - σ
s i-1,j –(ρ+σ)
Continue Gap in w (deletion)
Start Gap in w (deletion): from middle
si,j =
max
s i,j-1 - σ
s i,j-1 –(ρ+σ)
Continue Gap in v (insertion)
si,j =
max
si-1,j-1 + δ (vi, wj) Match or Mismatch
s i,j
End deletion: from top
s i,j
End insertion: from bottom
Start Gap in v (insertion):from middle
An Introduction to Bioinformatics
y1 ………………………… yN
Bounded Dynamic Programming
x1 ………………………… xM
Initialization:
F(i,0), F(0,j) undefined for i, j > k
Iteration:
For i = 1…M
For j = max(1, i – k)…min(N, i+k)
F(i – 1, j – 1)+ s(xi, yj)
F(i, j) = max F(i, j – 1) – d, if j > i – k(N)
F(i – 1, j) – d, if j < i + k(N)
k(N)
Termination:
same
Easy to extend to the affine gap case
An Introduction to Bioinformatics
Computing Alignment Path Requires
Quadratic Memory
Alignment Path
• Space complexity for
computing alignment path
for sequences of length n
and m is O(nm)
• We need to keep all
backtracking references in
memory to reconstruct the
path (backtracking)
m
n
An Introduction to Bioinformatics
Computing Alignment Score with Linear
Memory
Alignment Score
• Space complexity of
computing just the score
itself is O(n)
• We only need the previous nn
column to calculate the
current column, and we
can then throw away that
previous column once
we’re done using it
2
An Introduction to Bioinformatics
Computing Alignment Score: Recycling Columns
Only two columns of scores are saved at any
given time
memory for column
1 is used to
calculate column 3
memory for column
2 is used to
calculate column 4
An Introduction to Bioinformatics
Crossing the Middle Line
m/2
Define
Prefix(i)
Suffix(i)
n
We want to calculate the longest
m path from (0,0) to (n,m) that
passes through (i,m/2) where i
ranges from 0 to n and
represents the i-th row
length(i)
as the length of the longest path
from (0,0) to (n,m) that passes
through vertex (i, m/2)
An Introduction to Bioinformatics
Crossing the Middle Line
m/2
m
Prefix(i)
Suffix(i)
n
Define (mid,m/2) as the vertex where the longest path crosses
the middle column.
length(mid) = optimal length = max0i n length(i)
An Introduction to Bioinformatics
Computing Prefix(i)
• prefix(i) is the length of the longest path from
(0,0) to (i,m/2)
• Compute prefix(i) by dynamic programming in
the left half of the matrix
store prefix(i) column
0
m/2
m
An Introduction to Bioinformatics
Computing Suffix(i)
• suffix(i) is the length of the longest path from (i,m/2) to (n,m)
• suffix(i) is the length of the longest path from (n,m) to (i,m/2)
with all edges reversed
• Compute suffix(i) by dynamic programming in the right half
of the “reversed” matrix
store suffix(i) column
0
m/2
m
An Introduction to Bioinformatics
Length(i) = Prefix(i) + Suffix(i)
• Add prefix(i) and suffix(i) to compute length(i):
• length(i)=prefix(i) + suffix(i)
• You now have a middle vertex of the maximum
path (i,m/2) as maximum of length(i)
0
middle point found
i
0
m/2
m
An Introduction to Bioinformatics
Finding the Middle Point
0
m/4
m/2
3m/4
m
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Finding the Middle Point again
0
m/4
m/2
3m/4
m
An Introduction to Bioinformatics
And Again
0
m/8
m/4
3m/8
m/2
5m/8
3m/4 7m/8 m
An Introduction to Bioinformatics
Time = Area: First Pass
• On first pass, the algorithm covers the entire
area
Area = nm
An Introduction to Bioinformatics
Time = Area: First Pass
• On first pass, the algorithm covers the entire
area
Area = nm
Computing Computing
prefix(i)
suffix(i)
An Introduction to Bioinformatics
Time = Area: Second Pass
• On second pass, the algorithm covers only
1/2 of the area
Area/2
An Introduction to Bioinformatics
Time = Area: Third Pass
• On third pass, only 1/4th is covered.
Area/4
An Introduction to Bioinformatics
Geometric Reduction At Each Iteration
1 + ½ + ¼ + ... + (½)k ≤ 2
•
Runtime: O(Area) = O(nm)
5th pass: 1/16
3rd pass: 1/4
first pass: 1
4th pass: 1/8
2nd pass: 1/2
An Introduction to Bioinformatics
Indexing-based Local
Aligners
BLAST, WU-BLAST, BlastZ, MegaBLAST,
BLAT, PatternHunter, ……
An Introduction to Bioinformatics
Some useful applications of alignments
• Given a newly discovered gene,
• Does it occur in other species?
• How fast does it evolve?
• Assume we try Smith-Waterman:
Our
new
gene
104
The entire genomic database
1010 - 1011
An Introduction to Bioinformatics
Some useful applications of alignments
• Given a newly sequenced organism,
• Which subregions align with other organisms?
•
Potential genes
•
Other biological characteristics
• Assume we try Smith-Waterman:
Our newly
sequenced
mammal
3109
The entire genomic database
1010 - 1011
An Introduction to Bioinformatics
Indexing-based local alignment
(BLAST- Basic Local Alignment Search Tool)
Main idea:
1.
Construct a dictionary of all the words in the
query
2.
Initiate a local alignment for each word
match between query and DB
query
DB
Running Time: O(MN)
However, orders of magnitude faster than
Smith-Waterman
An Introduction to Bioinformatics
Indexing-based local alignment
Dictionary:
All words of length k (~10)
Alignment initiated between
words of alignment score  T
(typically T = k)
……
query
……
Alignment:
Ungapped extensions until score
below statistical threshold
scan
DB
Output:
All local alignments with score
> statistical threshold
query
An Introduction to Bioinformatics
Indexing-based local alignment—
A C G A A G T A A G G T
Extensions
k=4
The matching word GGTC
initiates an alignment
Extension to the left and
right with no gaps until
alignment falls < T
below best so far
Output:
GTAAGGTCC
GTTAGGTCC
C C C T T C C T G G A T T G C G A
Example:
C C A G T
An Introduction to Bioinformatics
•
Extensions with gaps
in a band around
anchor
Output:
GTAAGGTCCAGT
GTTAGGTC-AGT
C T G A T C C T G G A T T G C G A
Indexing-based local alignment—
A C G A A G T A A G G T C
Extensions
Gapped extensions
C A G T
An Introduction to Bioinformatics
Gapped extensions
until threshold
•
Extensions with gaps
until score < T below
best score so far
Output:
GTAAGGTCCAGT
GTTAGGTC-AGT
C T G A T C C T G G A T T G C G A
Indexing-based local alignment—
Extensions
A C G A A G T A A G G T
C C A G T
An Introduction to Bioinformatics
Indexing-based local alignment—The
index
• Sensitivity/speed
tradeoff
long words
(k = 15)

Sensitivity
Speed
short words
(k = 7)

Sens.
Speed
Kent WJ, Genome Research 2002
An Introduction to Bioinformatics
Indexing-based local alignment—The
index
Methods to improve sensitivity/speed
1. Using pairs of words
……ATAACGGACGACTGATTACACTGATTCTTAC……
……GGCACGGACCAGTGACTACTCTGATTCCCAG……
2. Using inexact words
……ATAACGGACGACTGATTACACTGATTCTTAC……
……GGCGCCGACGAGTGATTACACAGATTGCCAG……
3. Patterns—non consecutive positions
TTTGATTACACAGAT
T G TT CAC G
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Measured improvement
Kent WJ, Genome Research 2002
An Introduction to Bioinformatics
Non-consecutive words—Patterns
Patterns increase the likelihood of at least one
match within a long conserved region
Consecutive Positions
Non-Consecutive Positions
6 common
5 common
7 common
3 common
On a 100-long 70% conserved region:
Consecutive
Expected # hits:
1.07
Prob[at least one hit]:
0.30
Non-consecutive
0.97
0.47
An Introduction to Bioinformatics
Advantage of Patterns
11 positions
11 positions
10 positions
An Introduction to Bioinformatics
Multiple patterns
TTTGATTACACAGAT
T G TT CAC G
T G T C CAG
TTGATT A G
•
K patterns
• Takes K times longer to scan
•
•
How long does it take
to search the query?
Patterns can complement one another
Computational problem:
•
•
Given: a model (prob distribution) for homology between two regions
Find: best set of K patterns that maximizes Prob(at least one match)
Buhler et al. RECOMB 2003
Sun & Buhler RECOMB 2004
An Introduction to Bioinformatics
Variants of BLAST
•
•
NCBI BLAST: search the universe http://www.ncbi.nlm.nih.gov/BLAST/
MEGABLAST:
•
Optimized to align very similar sequences
•
•
•
WU-BLAST: (Wash U BLAST)
•
•
•
Uses inexact k-mers, sensitive
PatternHunter
•
•
Faster, less sensitive than BLAST
Good for aligning huge numbers of queries
CHAOS
•
•
Very good optimizations
Good set of features & command line arguments
BLAT
•
•
•
Works best when k = 4i  16
Linear gap penalty
Uses patterns instead of k-mers
BlastZ
•
Uses patterns, good for finding genes
An Introduction to Bioinformatics
Example
Query: gattacaccccgattacaccccgattaca (29 letters)
[2 mins]
Database: All GenBank+EMBL+DDBJ+PDB sequences (but no EST, STS, GSS, or phase
0, 1 or 2 HTGS sequences) 1,726,556 sequences; 8,074,398,388 total
letters
>gi|28570323|gb|AC108906.9| Oryza sativa chromosome 3 BAC OSJNBa0087C10 genomic
sequence, complete sequence Length = 144487 Score = 34.2 bits (17),
Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus
Query:
Sbjct:
4
tacaccccgattacaccccga 24
||||||| |||||||||||||
125138 tacacccagattacaccccga 125158
Score = 34.2 bits (17),
Expect = 4.5 Identities = 20/21 (95%) Strand = Plus / Plus
Query:
Sbjct:
4 tacaccccgattacaccccga 24
||||||| |||||||||||||
125104 tacacccagattacaccccga 125124
>gi|28173089|gb|AC104321.7|
Oryza sativa chromosome 3 BAC OSJNBa0052F07 genomic sequence, complete
sequence Length = 139823 Score = 34.2 bits (17), Expect = 4.5 Identities = 20/21 (95%) Strand =
Plus / Plus
Query:
Sbjct:
4 tacaccccgattacaccccga 24
||||||| |||||||||||||
3891 tacacccagattacaccccga 3911
An Introduction to Bioinformatics
Example
Query: Human atoh enhancer, 179 letters
[1.5 min]
Result: 57 blast hits
1.
2.
3.
4.
5.
6.
7.
8.
gi|7677270|gb|AF218259.1|AF218259 Homo sapiens ATOH1 enhanc...
gi|22779500|gb|AC091158.11| Mus musculus Strain C57BL6/J ch...
gi|7677269|gb|AF218258.1|AF218258 Mus musculus Atoh1 enhanc...
gi|28875397|gb|AF467292.1| Gallus gallus CATH1 (CATH1) gene...
gi|27550980|emb|AL807792.6| Zebrafish DNA sequence from clo...
gi|22002129|gb|AC092389.4| Oryza sativa chromosome 10 BAC O...
gi|22094122|ref|NM_013676.1| Mus musculus suppressor of Ty ...
gi|13938031|gb|BC007132.1| Mus musculus, Similar to suppres...
355 1e-95
264 4e-68
256 9e-66
78 5e-12
54 7e-05
44 0.068
42 0.27
42 0.27
gi|7677269|gb|AF218258.1|AF218258 Mus musculus Atoh1 enhancer sequence Length = 1517
Score = 256 bits (129), Expect = 9e-66 Identities = 167/177 (94%),
Gaps = 2/177 (1%) Strand = Plus / Plus
Query:
3 tgacaatagagggtctggcagaggctcctggccgcggtgcggagcgtctggagcggagca 62
||||||||||||| ||||||||||||||||||| ||||||||||||||||||||||||||
Sbjct: 1144 tgacaatagaggggctggcagaggctcctggccccggtgcggagcgtctggagcggagca 1203
Query:
63 cgcgctgtcagctggtgagcgcactctcctttcaggcagctccccggggagctgtgcggc 122
|||||||||||||||||||||||||| ||||||||| |||||||||||||||| |||||
Sbjct: 1204 cgcgctgtcagctggtgagcgcactc-gctttcaggccgctccccggggagctgagcggc 1262
Query:
123 cacatttaacaccatcatcacccctccccggcctcctcaacctcggcctcctcctcg 179
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Sbjct: 1263 cacatttaacaccgtcgtca-ccctccccggcctcctcaacatcggcctcctcctcg 1318